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SOLUTION: Since all the angles are congruent, the triangle is equiangular. 4-1 Classifying Triangles ARCHITECTURE Classify each triangle as acute, equiangular, obtuse, or right. Classify each triangle as acute, equiangular, obtuse, or right. Explain your reasoning. 1. Refer to the figure on page 240. 4. SOLUTION: is equiangular, since all three angles are congruent. SOLUTION: One angle of the triangle measures 90, so it is a right angle. Since the triangle has a right angle, it is a right triangle. 5. SOLUTION: In , Since the triangle obtuse. 2. Refer to the figure on page 240. . So, is obtuse. has an obtuse angle, it is 6. SOLUTION: In the figure, So by substitution, is a right triangle, since . . CCSS PRECISION Classify each triangle as equilateral, isosceles, or scalene. 7. SOLUTION: One angle of the triangle measures 120, so it is an obtuse angle. Since the triangle has an obtuse angle, it is an obtuse triangle. SOLUTION: The triangle has two congruent sides. So, it is isosceles. 3. Refer to the figure on page 240. 8. SOLUTION: No two sides are congruent in the given triangle. So, it is scalene. SOLUTION: Since all the angles are congruent, the triangle is equiangular. Classify each triangle as acute, equiangular, reasoning. eSolutions Manual Powered by Cognero obtuse, or -right. Explain your If point K is the midpoint of , classify each triangle in the figure as equilateral, isosceles, or scalene. Page 1 8. SOLUTION: 4-1 Classifying No two sidesTriangles are congruent in the given triangle. So, it is scalene. Also, HL = 3 and FL = 7. No two sides are congruent in is scalene. . Therefore, it ALGEBRA Find x and the measures of the unknown sides of each triangle. If point K is the midpoint of , classify each triangle in the figure as equilateral, isosceles, or scalene. 12. SOLUTION: In the figure, So, Solve for x. 9. SOLUTION: In , K is the midpoint of . So, . So, all the sides of have equal lengths. Therefore, is equilateral. Substitute in LN and MN. 10. SOLUTION: In , GL = GH + HL by Segment Addition Postulate. has two congruent sides. So, it is isosceles. 11. SOLUTION: In , K is the midpoint of . So, . 13. SOLUTION: Also, HL = 3 and FL = 7. No two sides are congruent in is scalene. . Therefore, it In the figure, So any combination of two side measures can be used to find x. ALGEBRA Find x and the measures of the unknown sides of each triangle. Substitute eSolutions Manual - Powered by Cognero 12. SOLUTION: in QR.. Page 2 Since all the sides are congruent, QR = RS = QS = 4; The total amount of wire needed, including the hook, is 2.1 + 3.2 + 3.2 + 1.5 or 10 cm. 45 cm ÷ 10 cm/earring = 4.5 earrings. There is not enough wire to make 5 earrings, only 4 can be made from 45 cm of wire. 4-1 Classifying Triangles Classify each triangle as acute, equiangular, obtuse, or right. 13. 15. SOLUTION: SOLUTION: One angle of the triangle measures 115, so it is a obtuse angle. Since the triangle has an obtuse angle, it is an obtuse triangle. In the figure, So any combination of two side measures can be used to find x. Substitute in QR.. 16. SOLUTION: The triangle has three acute angles that are not all equal. It is an acute triangle. Since all the sides are congruent, QR = RS = QS = 25. 14. JEWELRY Suppose you are bending stainless steel wire to make the earring shown. The triangular portion of the earring is an isosceles triangle. If 1.5 centimeters are needed to make the hook portion of the earring, how many earrings can be made from 45 centimeters of wire? Explain your reasoning. 17. SOLUTION: One angle of the triangle measures 90, so it is a right angle. Since the triangle has a right angle, it is a right triangle. 18. SOLUTION: Since all the angles are congruent, it is a equiangular triangle. SOLUTION: 4; The total amount of wire needed, including the hook, is 2.1 + 3.2 + 3.2 + 1.5 or 10 cm. 45 cm ÷ 10 cm/earring = 4.5 earrings. There is not enough wire to make 5 earrings, only 4 can be made from 45 cm of wire. 19. SOLUTION: The triangle has three acute angles. It is an acute triangle. Classify each triangle as acute, equiangular, obtuse, or right. 20. 15. eSolutions Manual - Powered by Cognero SOLUTION: One angle of the triangle measures 115, so it is a Page 3 SOLUTION: One angle of the triangle measures 90, so it is a right angle. Since the triangle has a right angle, it is a right 23. 19. SOLUTION: 4-1 Classifying The triangleTriangles has three acute angles. It is an acute triangle. SOLUTION: All the angles are acute angles in acute triangle. . So, it is an SOLUTION: All the angles are acute angles in acute triangle. . So, it is 24. 20. SOLUTION: One angle of the triangle measures 90, so it is a right angle. Since the triangle has a right angle, it is a right triangle. 25. SOLUTION: In , . So, is a right angle. Since the triangle has a right angle, it is a right triangle. CCSS PRECISION Classify each triangle as acute, equiangular, obtuse, or right. 26. SOLUTION: All the angles are congruent in equiangular. . So, it is Classify each triangle as equilateral, isosceles, or scalene. 27. Refer to the figure on page 241. 21. SOLUTION: In , Since the triangle obtuse. . So, is obtuse. has an obtuse angle, it is SOLUTION: Since all the sides are congruent, the triangle is equilateral. 22. SOLUTION: In , Since the triangle right triangle. 28. Refer to the figure on page 241. . So, is a right angle. has a right angle, it is a 23. SOLUTION: All the angles are acute angles in acute triangle. . So, it is an 24. SOLUTION: All the angles are acute angles in acute triangle. eSolutions Manual - Powered by Cognero 25. SOLUTION: SOLUTION: The triangle has two congruent sides. So, it is isosceles. 29. Refer to the figure on page 241. . So, it is Page 4 Here, E is the midpoint of . So, DE = EF = 5. Also by the Segment Addition Postulate, DF = DE + EF = 10. In , DF = 10 and AF = 10. The triangle has two congruent sides. So, it is isosceles. SOLUTION: 4-1 Classifying The triangle Triangles has two congruent sides. So, it is isosceles. 29. Refer to the figure on page 241. 33. SOLUTION: No two sides are congruent in scalene. . So, it is SOLUTION: No two sides are congruent in scalene. . So, it is 34. SOLUTION: No two sides are congruent in the given triangle, so it is scalene. If point C is the midpoint of and point E is the midpoint of , classify each triangle as equilateral, isosceles, or scalene. 35. SOLUTION: Here, C is the midpoint of . So, BC = CD = 4. Also by Segment Addition Postulate, BC = BC + CD = 8. In , AB = 8, BD = 8, and AD = 8. All the sides are congruent in . So, it is equilateral. 36. ALGEBRA Find x and the length of each side if is an isosceles triangle with 30. SOLUTION: In , all the sides are of different lengths. So, it is scalene. SOLUTION: 31. SOLUTION: Here, E is the midpoint of . So, DE = EF = 5. In , all the sides are having different lengths. So, it is scalene. Here, By the definition of congruence, Substitute. . 32. SOLUTION: Here, E is the midpoint of . So, DE = EF = 5. Also by the Segment Addition Postulate, DF = DE + EF = 10. In , DF = 10 and AF = 10. The triangle has two congruent sides. So, it is isosceles. 33. Substitute in AB, BC, and CA. SOLUTION: No two sides are congruent in scalene. . So, it is eSolutions Manual - Powered by Cognero Page 5 34. Here, C is the midpoint of . So, BC = CD = 4. Also by Segment Addition Postulate, BC = BC + CD = 8. In , AB = 8, BD = 8, and AD = 8. All the sides 4-1 Classifying Triangles are congruent in . So, it is equilateral. 36. ALGEBRA Find x and the length of each side if is an isosceles triangle with SOLUTION: Here, By the definition of congruence, Substitute. . 37. ALGEBRA Find x and the length of each side if is an equilateral triangle. SOLUTION: Since is an equilateral triangle, So any combination of sides can be used to find x. Let's use Solve for x. Substitute in AB, BC, and CA. Substitute in FG. Since all the sides are congruent, FG = GH = HF = 19. 38. GRAPHIC ART Classify each numbered triangle in Kat by its angles and by its sides. Use the corner of a sheet of notebook paper to classify angle measures and a ruler to measure sides. Refer to the figure on page 242. SOLUTION: 37. ALGEBRA Find x and the length of each side if is an equilateral triangle. SOLUTION: Since is an equilateral triangle, So any combination of eSolutions - Powered Cognero sidesManual can be used tobyfind x. Let's use Solve for x. 1: right scalene; this has 1 right angle and the lengths of each side are different 2: right scalene; this has 1 right angle and the lengths of each side are different 3: obtuse scalene; this has 1 obtuse angle and the lengths of each side are different 4: acute scalene; all angles are less than 90 and the lengths of each side are different 5: right scalene; this has 1 right angle and the lengths of each side are different 6: obtuse scalene; this has 1 obtuse angle and the lengths of each side are different 39. KALEIDOSCOPE Josh is building a kaleidoscope using PVC pipe, cardboard, bits of colored paper,Page and6 a 12-inch square mirror tile. The mirror tile is to be cut into strips and arranged to form an open prism with a base like that of an equilateral triangle. Make the lengths of each side are different 5: right scalene; this has 1 right angle and the lengths of each side are different 6: obtuse Triangles 4-1 Classifying scalene; this has 1 obtuse angle and the lengths of each side are different 39. KALEIDOSCOPE Josh is building a kaleidoscope using PVC pipe, cardboard, bits of colored paper, and a 12-inch square mirror tile. The mirror tile is to be cut into strips and arranged to form an open prism with a base like that of an equilateral triangle. Make a sketch of the prism, giving its dimensions. Explain your reasoning. 42. SOLUTION: No two sides are congruent in and it has a right angle. So, it is a scalene right triangle. COORDINATE GEOMETRY Find the measures of the sides of and classify each triangle by its sides. 43. X(–5, 9), Y(2, 1), Z(–8, 3) SOLUTION: Graph the points on a coordinate plane. SOLUTION: Because the base of the prism formed is an equilateral triangle, the mirror tile must be cut into three strips of congruent width. Since the original tile is a 12-inch square, each strip will be 12 inches long by 12 ÷ 3 or 4 inches wide. Use the Distance Formula to find the lengths of . CCSS PRECISION Classify each triangle in the figure by its angles and sides. has endpoints X(–5, 9) and Y(2, 1). Substitute. 40. SOLUTION: has two congruent sides and a right angle. So, it is an isosceles right triangle. 41. has endpoints Y(2, 1) and Z(–8, 3). Substitute. SOLUTION: In the figure, . has two congruent sides and an obtuse angle. So, it is an isosceles obtuse triangle. 42. SOLUTION: No two sides are congruent in and it has a right angle. So, it is a scalene right triangle. COORDINATE GEOMETRY Find the measures of the sides of and classify each triangle by its sides. 43. X(–5, 9), Y(2, 1), Z(–8, 3) eSolutions Manual - Powered by Cognero has endpoints X(–5, 9) and Z(–8, 3). Substitute. Page 7 has endpoints X(–5, 9) and Z(–8, 3). 4-1 Classifying Triangles Substitute. has endpoints X(7, 6) and Z(9, 1). Substitute. No two sides are congruent. So, it is scalene. 44. X(7, 6), Y(5, 1), Z(9, 1) SOLUTION: Graph the points on a coordinate plane. XY = XZ. This triangle has two congruent sides. So, it is isosceles. 45. X(3, –2), Y(1, –4), Z(3, –4) SOLUTION: Graph the points on a coordinate plane. Use the Distance Formula to find the lengths of . has end points X(7, 6) and Y(5, 1). Substitute. Use the Distance Formula to find the lengths of . has end points X(3, –2) and Y(1, –4). Substitute. has endpoints Y(5, 1) and Z(9, 1). Substitute. has endpoints Y(1, –4) and Z(3, –4). Substitute. has endpoints X(7, 6) and Z(9, 1). eSolutions Manual - Powered by Cognero Substitute. has endpoints X(3, –2) and Z(3, –4). Page 8 4-1 Classifying Triangles XY = XZ. This triangle has two congruent sides. So, it is isosceles. 45. X(3, –2), Y(1, –4), Z(3, –4) YZ = XZ = 2. This triangle has two congruent sides. So, it is isosceles. 46. X(–4, –2), Y(–3, 7), Z(4, –2) SOLUTION: Graph the points on a coordinate plane. SOLUTION: Plot the points on a coordinate plane. Use the Distance Formula to find the lengths of . Use the Distance Formula to find the lengths of . has endpoints X(–4, –2) and Y(–3, 7). has end points X(3, –2) and Y(1, –4). Substitute. Substitute. has endpoints Y(–3, 7) and Z(4, –2). has endpoints Y(1, –4) and Z(3, –4). Substitute. Substitute. has endpoints X(3, –2) and Z(3, –4). Substitute. has endpoints X(–4, –2) and Z(4, –2). Substitute. YZ = XZ = 2. This triangle has two congruent sides. So, it is isosceles. 46. X(–4, –2), -Y(–3, 7), by Z(4, –2) eSolutions Manual Powered Cognero SOLUTION: Plot the points on a coordinate plane. Page 9 No two sides are congruent. So, it is scalene. 47. PROOF Write a paragraph proof to prove that because because has endpoints X(–4, –2) and Z(4, –2). 4-1 Classifying Triangles Substitute. . We already know that is acute is acute. must also be acute is acute and . is acute by definition. 48. PROOF Write a two-column proof to prove that is equiangular if is equiangular and No two sides are congruent. So, it is scalene. 47. PROOF Write a paragraph proof to prove that is an acute triangle if and is acute. SOLUTION: Given: Prove: is acute. Proof: and form a linear pair. and are supplementary because if two angles form a linear pair, then they are supplementary. So, . We know , so by substitution, . Subtract to find that . We already know that is acute because is acute. must also be acute because is acute and . is acute by definition. SOLUTION: Given: is equiangular and Prove: is equiangular. Proof: Statements (Reasons) 1. is equiangular and (Given) 2. (Def. of equiangular ) 3. and (Corr. Post.) 4. (Substitution) 5. is equiangular. (Def. of equiangular ) ALGEBRA For each triangle, find x and the measure of each side. 49. is an equilateral triangle with FG = 3x – 10, GH = 2x + 5, and HF = x + 20. SOLUTION: Since is equilateral, FG = GH = HF. Consider FG = GH. 3x – 10 = 2x + 5 3x – 10 – 2x = 2x + 5 – 2x x –10 = 5 x =15 Substitute x =15. 48. PROOF Write a two-column proof to prove that is equiangular if is equiangular and Since all the sides are congruent, FG = GH = HF = 35. SOLUTION: Given: is equiangular and Prove: is equiangular. Proof: Statements (Reasons) 1. is equiangular and (Given) 2. (Def. of equiangular ) 3. and (Corr. Post.) 4. (Substitution) eSolutions Manual - Powered by Cognero 5. is equiangular. (Def. of equiangular ) ALGEBRA For each triangle, find x and the 50. is isosceles with 2x + 5, and LJ = 2x – 1. , JK = 4x – 1, KL = SOLUTION: Here, . By the definition of congruence, Substitute. . Page 10 Since all the sides are congruent, FG = GH = HF = 4-1 Classifying Triangles 35. 50. is isosceles with 2x + 5, and LJ = 2x – 1. , JK = 4x – 1, KL = 51. SOLUTION: Here, . By the definition of congruence, Substitute. Substitute is isosceles with . MN is two less than five times x, NP is seven more than two times x, and PM is two more than three times x. SOLUTION: Here, . By the definition of congruence, MN = 5x – 2, NP = 2x + 7, PM = 3x + 2 Substitute. . . in JK, KL, and LJ. Substitute in MN, NP, and PM. 51. is isosceles with . MN is two less than five times x, NP is seven more than two times x, and PM is two more than three times x. SOLUTION: Here, . By the definition of congruence, MN = 5x – 2, NP = 2x + 7, PM = 3x + 2 Substitute. Manual - Powered by Cognero eSolutions Substitute in MN, NP, and PM. . 52. is equilateral. RS is three more than four times x, ST is seven more than two times x, and TR is one more than five times x. SOLUTION: Since is equilateral, RS = ST = TR. RS = 4x + 3, ST = 2x + 7, TR = 5x + 1 Substitute x =2. Page 11 4-1 Classifying Triangles 52. is equilateral. RS is three more than four times x, ST is seven more than two times x, and TR is one more than five times x. SOLUTION: Since is equilateral, RS = ST = TR. RS = 4x + 3, ST = 2x + 7, TR = 5x + 1 Since all sides have the same length, they are all congruent. Therefore the triangle is equilateral. was constructed using AB as the length of each side. Since the arc for each segment is the same, the triangle is equilateral. 54. STOCKS Technical analysts use charts to identify patterns that can suggest future activity in stock prices. Symmetrical triangle charts are most useful when the fluctuation in the price of a stock is decreasing over time. a. Classify by its sides and angles the triangle formed if a vertical line is drawn at any point on the graph. b. How would the price have to fluctuate in order for the data to form an obtuse triangle? Draw an example to support your reasoning. Substitute x =2. Since all the sides are congruent, RS = ST = TR = 11. 53. CONSTRUCTION Construct an equilateral triangle. Verify your construction using measurement and justify it using mathematics. (Hint: Use the construction for copying a segment.) SOLUTION: SOLUTION: a. The two lines represent the highs and lows of a stock converging on the same price. Since the triangle is symmetrical, the top side would be a reflection of the bottom side. So, the two segments are congruent and the triangle is isosceles.The base angles of an isosceles triangle would be congruent acute angles and the angle formed by the converging prices is acute. Therefore, the triangle is acute. b. Sample answer: The fluctuation (the difference between the high and low prices of the stock) would have to be high and decrease quickly (over a short period of time) in order to form an obtuse triangle. A greater fluctuation will stretch out the angle, making its measure increase. Sample answer: In AB = BC = AC = 1.3 cm. Since all sides have the same length, they are all congruent. Therefore the triangle is equilateral. was constructed using AB as the length of each side. Since the arc for each segment is the same, the triangle is equilateral. 54. STOCKS Technical analysts use charts to identify patterns that can suggest future activity in stock prices. Symmetrical triangle charts are most useful when the fluctuation in the price of a stock is decreasing over time. a. Classify by its sides and angles the triangle formed if a vertical line is drawn at any point on the graph. b. How would the price have to fluctuate in order for the data to -form an by obtuse triangle? Draw an eSolutions Manual Powered Cognero example to support your reasoning. 55. MULTIPLE REPRESENTATIONS In the diagram, the vertex opposite side is a. GEOMETRIC Draw four isosceles triangles, including one acute, one right, and one obtuse Page 12 isosceles triangle. Label the vertices opposite the congruent sides as A and C. Label the remaining vertex B. Then measure the angles of each triangle 55. MULTIPLE REPRESENTATIONS In the 4-1 Classifying Triangles diagram, the vertex opposite side is a. GEOMETRIC Draw four isosceles triangles, including one acute, one right, and one obtuse isosceles triangle. Label the vertices opposite the congruent sides as A and C. Label the remaining vertex B. Then measure the angles of each triangle and label each angle with its measure. b. TABULAR Measure all the angles of each triangle. Organize the measures for each triangle into a table. Include a column in your table to record the sum of these measures. c. VERBAL Make a conjecture about the measures of the angles that are opposite the congruent sides of an isosceles triangle. Then make a conjecture about the sum of the measures of the angles of an isosceles triangle. d. ALGEBRAIC If x is the measure of one of the angles opposite one of the congruent sides in an isosceles triangle, write expressions for the measures of each of the other two angles in the triangle. Explain. SOLUTION: a. Sample answer: b. c. Sample answer: In an isosceles triangle, the angles opposite the congruent sides have the same measure. The sum of the measures of the angles of an isosceles triangle is 180. d. x and 180 – 2x; If the measures of the angles opposite the congruent sides of an isosceles triangle have the same measure, then if one angle measures x, then the other angle also measures x. The sum of the measures of the angles of an isosceles triangle is 180, thus the measure of the third angle is 180 – 2x. 56. ERROR ANALYSIS Elaina says that is obtuse. Ines disagrees, explaining that the triangle has more acute angles than obtuse angles so it must be acute. Is either of them correct? Explain your reasoning. eSolutions Manual - Powered by Cognero SOLUTION: Sample answer: Elaina; all triangles have at least two acute angles, so using Ines’ reasoning all triangles would be classified as acute. Instead, triangles are classified by their third angle. If the third angle is also acute, then the triangle is acute. If the third angle is obtuse, as in the triangle shown, the triangle is Page 13 classified as obtuse. CCSS PRECISION Determine whether the opposite the congruent sides of an isosceles triangle have the same measure, then if one angle measures x, then the other angle also measures x. The sum of the measuresTriangles of the angles of an isosceles triangle is 4-1 Classifying 180, thus the measure of the third angle is 180 – 2x. 56. ERROR ANALYSIS Elaina says that is obtuse. Ines disagrees, explaining that the triangle has more acute angles than obtuse angles so it must be acute. Is either of them correct? Explain your reasoning. SOLUTION: Sample answer: Elaina; all triangles have at least two acute angles, so using Ines’ reasoning all triangles would be classified as acute. Instead, triangles are classified by their third angle. If the third angle is also acute, then the triangle is acute. If the third angle is obtuse, as in the triangle shown, the triangle is classified as obtuse. SOLUTION: Never; all equilateral triangles are also equiangular, which means all of the angles are 60°. A right triangle has one 90° angle. 60. CHALLENGE An equilateral triangle has sides that measure 5x + 3 units and 7x – 5 units. What is the perimeter of the triangle? Explain. SOLUTION: Sample answer: Since the triangle is equilateral, the sides are equal. Setting 5x + 3 equal to 7x – 5 and solving, x is 4. The length of one side is 5(4) + 3 or 23 units. The perimeter of an equilateral triangle is the sum of the three sides or three times one side. The perimeter is 3(23) or 69 units. OPEN ENDED Draw an example of each type of triangle below using a protractor and a ruler. Label the sides and angles of each triangle with their measures. If not possible, explain why not. 61. scalene right SOLUTION: Sample answer: CCSS PRECISION Determine whether the statements below are sometimes, always, or never true. Explain your reasoning. 57. Equiangular triangles are also right triangles. SOLUTION: Never; all equiangular triangles have three 60° angles, so they do not have a 90° angle. Therefore they cannot be right triangles. 58. Equilateral triangles are isosceles. 62. isosceles obtuse SOLUTION: Sample answer: SOLUTION: Always; all equilateral triangles have three equal sides and isosceles triangles have at least two equal sides, so all triangles with three equal sides are isosceles. 59. Right triangles are equilateral. SOLUTION: Never; all equilateral triangles are also equiangular, which means all of the angles are 60°. A right triangle has one 90° angle. 60. CHALLENGE An equilateral triangle has sides that measure 5x + 3 units and 7x – 5 units. What is the perimeter of the triangle? Explain. SOLUTION: Sample answer: Since the triangle is equilateral, the sides are equal. Setting 5x + 3 equal to 7x – 5 and solving, x is 4. The length of one side is 5(4) + 3 or 23 units. The perimeter of an equilateral triangle is eSolutions Manual Powered Cognero the sum of-the threebysides or three times one side. The perimeter is 3(23) or 69 units. 63. equilateral obtuse SOLUTION: Not possible; all equilateral triangles have three acute angles. 64. WRITING IN MATH Explain why classifying an equiangular triangle as an acute equiangular triangle is unnecessary. SOLUTION: Sample answer: An acute triangle has three acute angles and an equiangular triangle has three angles that measure 60°. Since an angle that measures 60° is an acute angle, all equiangular triangles are acute. Therefore, acute equiangular is redundant. Page 14 65. Which type of triangle can serve as a counterexample to the conjecture below? 63. equilateral obtuse SOLUTION: Not possible;Triangles all equilateral triangles have three acute 4-1 Classifying angles. 64. WRITING IN MATH Explain why classifying an equiangular triangle as an acute equiangular triangle is unnecessary. SOLUTION: Sample answer: An acute triangle has three acute angles and an equiangular triangle has three angles that measure 60°. Since an angle that measures 60° is an acute angle, all equiangular triangles are acute. Therefore, acute equiangular is redundant. So, $33.80 was deducted from the original price. The correct option is H. 67. GRIDDED RESPONSE Jorge is training for a 20mile race. Jorge runs 7 miles on Monday, Tuesday, and Friday, and 12 miles on Wednesday and Saturday. After 6 weeks of training, Jorge will have run the equivalent of how many races? SOLUTION: Jorge runs 7 miles on Monday, Tuesday, and Friday, and 12 miles on Wednesday and Saturday. Total miles per week = 7 + 7 + 7 +12 + 12 = 45 miles He will run 45 times 6 or 270 miles in 6 weeks. Number of races = 270/20 =13.5 65. Which type of triangle can serve as a counterexample to the conjecture below? 68. SAT/ACT What is the slope of the line determined by the equation 2x + y = 5? A A equilateral B obtuse C right D scalene B –2 C –1 D 2 E SOLUTION: An acute angle is less than 90. In an equilateral triangle, all 3 angles are acute. Therefore, an equilateral triangle is a counterexample to the given statement. The correct answer is A. 66. ALGEBRA A baseball glove originally cost $84.50. Kenji bought it at 40% off. How much was deducted from the original price? F $50.70 G $44.50 H $33.80 J $32.62 SOLUTION: Write the equation 2x + y = 5 in slope-intercept form. y = –2x + 5 Here, slope is –2. So, the correct option is B. Find the distance between each pair of parallel lines with the given equations. 69. SOLUTION: The two lines are of the form x = a. So, the slopes are undefined. Therefore, the lines are vertical lines passing through x = –2 and x = 5 respectively. The perpendicular distance between the two vertical lines is 5 – (–2) = 7 units. SOLUTION: We need to find 40% of 84.50 in order to calculate the discount. 70. So, $33.80 was deducted from the original price. The correct option is H. SOLUTION: The two lines have the coefficient of x, zero. So, the slopes are zero. Therefore, the lines are horizontal lines passing through y = –6 and y = 1 respectively. The perpendicular distance between the two horizontal lines is 1 – (–6) = 7 units. 67. GRIDDED RESPONSE Jorge is training for a 20mile race. Jorge runs 7 miles on Monday, Tuesday, and Friday, and 12 miles on Wednesday and Saturday. After 6 weeks of training, Jorge will have run the equivalent of how many races? SOLUTION: JorgeManual runs -7Powered miles on eSolutions by Monday, Cognero Tuesday, and Friday, and 12 miles on Wednesday and Saturday. Total miles per week = 7 + 7 + 7 +12 + 12 = 45 miles 71. Page 15 SOLUTION: The slope of a line perpendicular to both the lines will The two lines have the coefficient of x, zero. So, the slopes are zero. Therefore, the lines are horizontal lines passing through y = –6 and y = 1 respectively. The perpendicular distance between the two 4-1 Classifying Triangles horizontal lines is 1 – (–6) = 7 units. 71. Therefore, the distance between the two lines is 72. SOLUTION: The slope of a line perpendicular to both the lines will be . Consider the y-intercept of any of the two lines and write the equation of the perpendicular line through it. The y-intercept of the line y = 2x – 7 is (0, –7). So, the equation of a line with slope y-intercept of –7 is and a . The perpendicular meets the line y = 2x – 7 at (0, – 7). To find the point of intersection of the perpendicular and the other line, solve the two equations. The left sides of the equations are the same. So, equate the right sides and solve for x. SOLUTION: The slope of a line perpendicular to both the lines will be –1. Consider the y-intercept of any of the two lines and write the equation of the perpendicular line through it. The y-intercept of the line y = x – 4 is (0, –4). So, the equation of a line with slope –1 and a yintercept of –4 is . The perpendicular meets the line y = x – 4 at (0, –4). To find the point of intersection of the perpendicular and the other line, solve the two equations. The left sides of the equations are the same. So, equate the right sides and solve for x. Use the value of x to find the value of y. Use the value of x to find the value of y. So, the point of intersection is (–4, –5). Use the Distance Formula to find the distance between the points (–4, –5) and (0, –7). Therefore, the distance between the two lines is 72. SOLUTION: The slope of a line perpendicular to both the lines will be –1. Consider theby y-intercept of any of the two eSolutions Manual - Powered Cognero lines and write the equation of the perpendicular line through it. The y-intercept of the line y = x – 4 is (0, –4). So, the equation of a line with slope –1 and a y- So, the point of intersection is (–3, –1). Use the Distance Formula to find the distance between the points (–3, –1) and (0, –4). Therefore, the distance between the two lines is 73. FOOTBALL When striping the practice football field, Mr. Hawkins first painted the sidelines. Next he marked off 10-yard increments on one sideline. He then constructed lines perpendicular to the sidelines at each 10-yard mark. Why does this guarantee that the 10-yard lines will be parallel? SOLUTION: The lines at each 10-yard mark are perpendicular to the sideline. Since two lines perpendicular to the same line are parallel, Mr. Hawkins can be certain that the 10-yard lines are parallel. Identify the hypothesis and conclusion of each conditional statement. Page 16 74. If three points lie on a line, then they are collinear. SOLUTION: SOLUTION: The lines at each 10-yard mark are perpendicular to the sideline. Since two lines perpendicular to the same line areTriangles parallel, Mr. Hawkins can be certain 4-1 Classifying that the 10-yard lines are parallel. 80. Name three points that are collinear. Identify the hypothesis and conclusion of each conditional statement. 74. If three points lie on a line, then they are collinear. 81. Are points D, E, C, and B coplanar? SOLUTION: Points E, F, and C lie in the same line. Thus they are collinear. SOLUTION: Points D, C, and B lie in plane N, but point E does not lie in plane N. Thus, they are not coplanar. SOLUTION: H: three points lie on a line; C: the points are collinear Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. 75. If you are a teenager, then you are at least 13 years old. SOLUTION: H: you are a teenager; C: you are at least 13 years old 76. If 2x + 6 = 10, then x = 2. SOLUTION: H: 2x + 6 = 10; C: x = 2 82. SOLUTION: Alternate interior angles 77. If you have a driver’s license, then you are at least 16 years old. SOLUTION: H: you have a driver’s license; C: you are at least 16 years old 83. SOLUTION: Consecutive interior angles Refer to the figure. 84. SOLUTION: Alternate interior angles 78. How many planes appear in this figure? SOLUTION: The planes in the figure are: Plane AEB, plane BEC, plane CED, plane AED, and plane N. So, there are 5 planes appear in this figure. 85. SOLUTION: Alternate exterior angles 79. Name the intersection of plane AEB with plane N. SOLUTION: Plane AEB intersects with plane N in 80. Name three points that are collinear. SOLUTION: Points E, F, and C lie in the same line. Thus they are collinear. 81. Are points D, E, C, and B coplanar? SOLUTION: Points D, C, and B lie in plane N, but point E does not eSolutions Manual - Powered by Cognero lie in plane N. Thus, they are not coplanar. Identify each pair of angles as alternate interior, Page 17